For our numerical experiments we decided to use the free software FreeFEM++ v 3.50 (see http://www.freefem.org/, see [1]}), complemented with the library NLopt (see http://ab-initio.mit.edu/wiki/index.php/NLopt) using the Method of Moving Asymptotes as the optimizing routing (see [2]}). This technique is a gradient method based on a spatial type of convex approximation where in each iteration a strictly convex approximation subproblem is generated and solved. For the implementation of this algorithm the main required data are the initialization \(V_0\) , the associated routines to the cost and volume function and the associated routines to the gradient of the cost and volume function using the adjoint state. The admissible potentials \(V\)  take values in \([0,+\infty ]\)  but from the numerical point of view it is advisable to constrain \(V\)  to take values on a bounded interval \([0,V_{max} ]\) , with \(V_{max}\)  large enough. These data are required for the algorithm too. We observe that, when \(V\)  takes its maximal value \(V_{max}\) , the state \(u\)  is very small and practically vanishes, according to the well-posed character of the extremal problem and the state equation. This is consistent with the necessary conditions of optimality obtained in Section .
